
Nature of the decay products.

At least another active neutrino mass eigenstate \(\nu _j\) .
Since energy and angular momentum conservation ensure the lightest neutrino state to be stable, the cosmological constraint on \(\sum m_\nu \)  cannot be relaxed by more than \( 0.06\,\mathrm {eV}\)  and \(0.1\,\mathrm {eV}\)  for NH and IH, respectively.
Although it can get very relevant in the future, as discussed before, we will not focus on this possibility, since the relaxation is tightly constraint.

Only BSM species.
In this case, the constraint on \(\sum m_\nu \)  can be significantly relaxed to the level of \(\sum m_\nu \le 1\,\mathrm {eV}\)  at 95% CL, provided that the BSM particles are massless.
This is frequently called neutrino decay into dark radiation.
The exact bound is subject to ongoing investigation and recently updated in Ref. [1]}.
New experimental data sets, including the latest Planck 2018+BAO+Pantheon, and a refined analysis significantly strengthened the bound on a possible cosmological neutrino decay while being non relativistic.
The main constraining power arises from the time modification of the gravitational pull and the resulting modification of the Hubble parameter and hence a modified late time large scale structure formation.
On the other hand, if \(1/\tau _\nu \ge H_0 \sqrt{\Omega _m} (\sum m_\nu /(9 T_{\nu ,0}))^{3/2}\) , neutrinos decay while being relativistic and the same reasoning can not be applied.
However, decays in such a regime can be constrained by noticing that decay and inverse decay processes alter the neutrino free-streaming properties to lower scales which results in a phase shift of the high multipole moments in the cosmic microwave background power spectrum.
The resulting bound in the phase space of active neutrino lifetime and its mass is shown in figure REF .


Number of particles in the final state.

2-body decays.
Angular momentum conservation requires the decay products to be one fermion and one boson with spin 0 (scalar \(\phi \) ) or 1 (vector \(Z^{\prime }\) ). Regarding the fermion, we will consider two possibilities: a light neutrino mass eigenstate \(\nu _i\)  or a sterile neutrino \(\nu _4\) .

3-body decays and beyond.
For sufficiently massive bosons, i.e. \(m_{\phi ,\,Z^{\prime }} > m_{\nu _i}\) , any 2-body decay is kinematically closed. However, such a boson can mediate off-shell a 3-body decay, which becomes the dominant channel.
In a model independent analysis only the decay of \(\nu _i \rightarrow \nu _4\nu _4\nu _4\)  allows to relax the cosmological neutrino mass bound in a broad region of parameter space [2]}.
However, in realistic neutrino models the mixing between active and sterile neutrino states will close the viable window completely.
Hence, we will not focus on 3-body decays and beyond here.


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